#7653743

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39469	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(10,14)(15,16)(17,18,23,28)(19,22,27,24)(20,26,21,25)(29,32,30,31);(1,2)(3,6)(4,7)(5,8)(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,32)(26,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,15,14,16)(17,20,23,21)(18,25,28,26)(19,29,27,30)(22,31,24,32);(17,21,23,20)(18,26,28,25)(19,30,27,29)(22,32,24,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,29)(21,30)(23,27)(24,28)(25,31)(26,32);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,16,14,15)(17,22)(18,27)(19,28)(20,31)(21,32)(23,24)(25,30)(26,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7560939);